How clean is bar soap in a public bathroom? Is it “self-cleaning,” since it’s soap? It seems like a health hazard to me.

What’s neglected to be considered here is that soap doesn’t exactly make the dirty stuff “disappear”. It’s main purpose is to loosen the dirty stuff from whatever it is that is being cleaned so that water can carry the dirty stuff away when rinsing off the soap suds. So, a bar of soap does not self-clean at all in the absence of something else (water in this case) to pick up and carry the dirty stuff away from it. A bar of soap dropped into mud continues to be covered in mud (read: dirty) until the mud is either rinsed or scraped off of the soap.